Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 568072, 19 pages
doi:10.1155/2011/568072
Research Article
On the Extensions of Krasnoselskii-Type Theorems
to p-Cyclic Self-Mappings in Banach Spaces
M. De la Sen
Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus de Bizkaia,
P.O. Box 644, 48080 Bilbao, Spain
Correspondence should be addressed to M. De la Sen, manuel.delasen@ehu.es
Received 4 March 2011; Revised 22 June 2011; Accepted 26 June 2011
Academic Editor: Zhen Jin
Copyright q 2011 M. De la Sen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A set of np≥2-cyclic and either continuous or contractive self-mappings, with at least one of
them being contractive, which are defined on a set of subsets of a Banach space, are considered to
build a composed self-mapping of interest. The existence and uniqueness of fixed points and the
existence of best proximity points, in the case that the subsets do not intersect, of such composed
mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii-type
theorems.
1. Introduction
In the last years, important attention is being devoted to extend the fixed point theory by
weakening the conditions on both the maps and the sets where those maps operate 1, 2.
For instance, every nonexpansive self-mappings on weakly compact subsets of a metric
space has fixed points if the weak fixed point property holds 1. It has also to be pointed
out the relevance of fixed point theory in the stability of complex continuous-time and
discrete-time dynamic systems 3–5. On the other hand, Meir-Keeler self-mappings have
received important attention in the context of fixed point theory perhaps due to the associated
relaxing in the required conditions for the existence of fixed points compared with the usual
contractive mappings 6–10. Another interest of such maps is their usefulness as formal tool
for the study of p-cyclic contractions even if the involved subsets of the metric space under
study do not intersect 6. The underlying idea is that the best proximity points are fixed
points if such subsets intersect while they play a close role to fixed points otherwise. It has
to be pointed out that there are close links between contractive self-mappings and Kannan
self-mappings 2, 10–13.
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A rich research is being devoted to the existence and uniqueness of best proximity
points of cyclic mappings under different assumptions on the vector space to which the
subsets involved in the cyclic mapping belong. For instance, in 14, the concept of 2-cyclic
self-mappings is extended to p≥2-cyclic self-mappings and results about fixed points are
derived in the case that the subsets of the considered complete metric space have a nonempty
intersection. Some of the ideas in such a manuscript inspired the definition of p-cyclic
self-mappings from X to X with X being the union of the subsets involved in the cyclic
representation. On the other hand, the existence and uniqueness of best proximity points
of p≥2-cyclic ϕ-contractive self-mappings are investigated in 15 borrowing the previous
scenario investigated in 16, 17 for 2-cyclic ϕ-contractive self-mappings. Generally speaking,
the so-called cyclic ϕ-contractive self-mappings, which are associated with some strictly
increasing unbounded map, are based on a concept of weak contractiveness contrarily to the
standard, and commonly used, strict contractive concept being inspired in the well-known
Banach contraction principle. In those papers, the Banach space X under consideration is also
assumed to be reflexive and strictly convex. These joint assumptions, which are less restrictive
than the assumption that the space is uniformly convex since uniformly convex Banach
spaces are reflexive and strictly convex but the converse is not true in general, are proven
to keep intact the essential properties of existence and uniqueness of best proximity points
for cyclic ϕ-contractive self-mappings if the involved subsets are nonempty, weakly closed,
and convex. The above formalism is revisited in 18 for cyclic ϕ-contractive self-mappings
in the framework of ordered metric spaces. In 19, characterization of best proximity points
is studied for non-self-mappings S, T : A → B, where A and B are nonempty subsets of a
metric space. In general, best proximity points do not fulfil in this context the usual condition
x Sx T x. However, they jointly globally optimize the mappings from x to the distances
dx, T x and dx, Sx.
In this manuscript, X, d is a complete metric space and is considered associated to a
Banach space X endowed with translation-invariant and homogeneous metric d : X × X →
R0 and SX : {Ai ⊆ X : Ajp ≡ Aj ; ∀i ∈ p, ∀j ∈ Z0 } is a set of p subsets of X. T : i∈p Ai →
i∈p Ai is a p-cyclic self-mapping if it satisfies T Ai ⊆ Ai1 ; for all i ∈ p. A valid metric
is the norm of the Banach space X. If the p-cyclic self-mapping T : i∈p Ai → i∈p Ai is
nonexpansive resp., contractive—also referred to as a 2-cyclic contraction then there exists
a real constant k ∈ 0, 1 resp., k ∈ 0, 1 such that
d T x, T y ≤ kd x, y 1 − k distA, B,
∀x ∈ A, ∀y ∈ B.
1.1
The self-mapping T : A ∪ B → A ∪ B is said to be a 2-cyclic large contraction if A and B are
two given intersecting subsets of X if
d T x, T y < d x, y , ∀x ∈ A, y /
x ∈ B,
∀ε ∈ R , ∀x ∈ A, y ∈ B : d T x, T y ≥ ε ⇒ ∃δ ∈ 0, 1 : d T x, T y ≤ δd x, y .
1.2
This concept of 2-cyclic large contraction on intersecting subsets extends that of large
contraction 20, and both concepts extend to p-cyclic contractions. A p-cyclic large
contraction satisfies also 1.2 by replacing A → Ai , B → Ai1 ; for all i ∈ p provided that
∅. In Section 2 of this paper, we consider a mapping T : A1 ∪ A2 × A1 ∪ A2 → X
i∈p Ai /
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j
j
defined by T x, y T1 x T2 y such that T j x, y T T j−1 x, y T1 x T2 y; for all j≥2
∈ Z ; for all x, y ∈ Ai , for all i ∈ 2 and then T x, y ∈ Ai1 where Ti : A1 ∪ A2 → A1 ∪ A2 ;
i 1, 2 are 2-cyclic self-mappings. It is not required for most of the obtained results that
T : A1 ∪ A2 × A1 ∪ A2 → A1 ∪ A2 so that such a map is not required to be 2-cyclic either.
For the obtained results related to boundedness of distances between iterates through T , it is
not required for the set of subsets of X to be either closed or convex. For the obtained results
concerning fixed points and best proximity point, the sets A1 and A2 are required to be convex
but they are not necessarily closed if the self-mapping T can be defined on the union of the
closures of the sets A1 and A2 . Also, concerning best proximity points in the case that the
subsets do not intersect, it is assumed that the space is restricted to be a uniformly convex
Banach space 6–9. It turns out that since uniformly convex Banach spaces are also strictly
convex, the results also hold under this more general assumption used in several papers see,
for instance 15–18. In Section 3, the results of Section 2 are extended to mappings built in
a close way via p≥2-cyclic self-mappings on a set of p subsets Ai i ∈ p of X.
1.1. Notation
R0 : R ∪ {0},
Z0 : Z ∪ {0},
p : 1, 2, . . . , p ⊂ Z .
1.3
Superscript denotes vector or matrix transpose, FixT is the set of fixed points of a selfmapping T on some nonempty convex closed subset A of a metric space X, d, cl A denotes
the closure of a subset A of X, DomT and ImT denote, respectively, the domain and image
of the self-mapping T and 2X is the family of subsets of X, distA, B dAB denotes the
distance between the sets A and B for a 2-cyclic self-mapping T : A ∪ B → A ∪ B what is
simplified as distAi , Ai1 dAi Ai1 di ; for all i ∈ p for distances between adjacent subsets
p
of p-cyclic self-mappings T on i1 Ai where Ai i ∈ p are subsets of X.
BPi T is the set of best proximity points on a subset Ai of a metric space X, d of a
p
p-cyclic self-mapping T on i1 Ai , the union of a collection of nonempty subsets of X, d
which do not intersect.
2. Results for Mappings Defined by 2-Cyclic Self-Mappings
The following result is concerned with the above-defined map T : A1 ∪ A2 × A1 ∪ A2 → X
constructed with nonexpansive or contractive 2-cyclic self-mappings Ti : A1 ∪ A2 → A1 ∪ A2
for i 1, 2.
Theorem 2.1. The following properties hold.
i If Ti : A1 ∪ A2 → A1 ∪ A2 are both 2-cyclic nonexpansive self-mappings then
j
j
j
j
d T j x, y , T j x , y ≤ d T1 x, T1 x d T2 y, T2 y ≤ d x, x d y, y
≤ diamA1 diamA2 distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z ,
2.1
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lim sup d T j x, y , T j x , y ≤ diamA1 diamA2 distA1 , A2 ,
j →∞
2.2
∀x, y ∈ A1 , ∀x , y ∈ A2 .
ii If Ti : A1 ∪ A2 → A1 ∪ A2 for i 1, 2 are both 2-cyclic nonexpansive self-mappings and
at least one of them is contractive then Property (i) holds and, furthermore,
d T j x, y , T j x , y ≤ maxdiamA1 , diamA2 1 − mink1 , k2 distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z ,
2.3
lim sup d T j x, y , T j x , y ≤ maxdiamA1 , diamA2 1 − mink1 , k2 distA1 , A2 ,
j →∞
∀x, y ∈ A1 , ∀x , y ∈ A2 .
2.4
iii If Ti : A1 ∪ A2 → A1 ∪ A2 for i 1, 2 are both 2-cyclic contractive self-mappings then
Properties (i), (ii) hold and, furthermore,
j j j
j
d T j x, y , T j x , y ≤ k1 d x, x k2 d y, y 2 − k1 − k2 distA1 , A2 ≤ k1 d x, x k2 d y, y 2 − k1 − k2 distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z ,
lim sup d T j x, y , T j x , y ≤ 2 distA1 , A2 ,
2.5a
2.5b
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.6
∃ lim d T j x, y , T j x , y 0 if distA1 , A2 0, ∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.7
j→∞
j→∞
(i.e., A1 and A2 are closed and intersect or at least one of them is open while their boundaries
intersect).
Proof. Take x, y ∈ A1 and x , y ∈ A2 . Direct calculation yields by taking into account that
Ti : A1 ∪ A2 → A1 ∪ A2 ; i 1, 2 are 2-cyclic, then both self-mappings satisfy Ti A1 ⊆ A2 ,
Ti A2 ⊆ A1 for i 1, 2, and 1.1 with respective contraction constants k1 and k2 , and that the
metric is translation-invariant and homogeneous:
d T x, y , T x , y d T1 x T2 y, T1 x T2 y
d T1 x, T1 x T2 y − T2 y
≤ d T1 x, T1 x d T1 x , T1 x T2 y − T2 y
≤ d T1 x, T1 x d T2 y, T2 y
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≤ k1 d x, x k2 d y, y 1 − k1 distA1 , A2 1 − k2 distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z .
2.8
If k1 k2 1 then one gets from 2.8
j
j
j
j
d T j x, y , T j x , y ≤ d T1 x, T1 x d T2 y, T2 y ,
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z
j
j j ≤ k1 d x, x k2 d y, y k1 1 − k1 distA1 , A2 1
k2 1
− k2 distA1 , A2 , ∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z .
2.9
If k1 k2 1; that is, if both 2-cyclic self-mappings Ti : A1 ∪ A2 → A1 ∪ A2 i 1, 2 are
nonexpansive. Thus, Property i follows from 2.9. If only one of them is nonexpansive and
the other is contractive then maxk1 , k2 1 and 0 ≤ mink1 , k2 < 1 so that 2.3 follow from
2.9 and Property ii is proven. For real constants for some real constants ki ∈ 0, 1, Ti :
A1 ∪ A2 → A1 ∪ A2 i 1, 2 are both contractive and one gets from 2.9
j j d T j x, y , T j x , y ≤ k1 d x, x k2 d y, y
j−1
k1 1 − k1 distA1 , A2 k2 1 − k2 distA1 , A2 ,
2.10
0
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z
what leads to 2.5a, 2.5b, and to 2.6, 2.7 by taking j → ∞. Property iii has been
proven.
Corollary 2.2. If Ti : A1 ∪A2 → A1 ∪A2 are both 2-cyclic asymptotically contractive self-mappings,
that is, they are nonexpansive with time-varying real sequences {kij }j∈Z in 0, 1 for i 1, 2 which
converge asymptotically to two respective real numbers k1 and k2 in 0, 1 as j → ∞. Assume that
the sequences {kij }j∈Z for i 1, 2 are such that
lim sup
j →∞
j j
1
k1i 1 − k1 i1
j
k2i 1 − k2 ≤K<∞
2.11
i1
is satisfied for some real K ∈ R . Then, 2.1 holds and, furthermore, the following properties hold:
i
d T j x, y , T j x , y ≤ 2εj K distA1 , A2 ≤ 2ε Kj distA1 , A2 ,
∀j ≥ j0 ∈ Z ,
∀x, y ∈ A1 , ∀x , y ∈ A2
2.12
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Discrete Dynamics in Nature and Society
for any bounded positive decreasing real sequence εj , with arbitrary prescribed upperbound ε, which converges asymptotically to zero as j → ∞ for some finite j0 j0 ε ∈ Z
and some bounded nondecreasing positive real sequence Kj of upper-bound K.
ii
lim sup d T j x, y , T j x , y ,
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
j →∞
2.13
∃ lim d T j x, y , T j x , y 0,
if distA1 , A2 0, ∀x, y ∈ A1 , ∀x , y ∈ A2 .
j→∞
Proof. Note that
d T j x, y , T j x , y ≤
j
k1 d x, x j
k2 d y, y
1
j
1
1
j
k1i 1−k1 i1
j
k2i 1−k2 distA1 , A2 ,
i1
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z .
2.14
Since the terms in the sum the summation term of the right-hand-side of the above equation
are all nonnegative, it follows that:
lim sup
j j
j →∞
1
≥
k1i 1 − k1 i1
j
k2i 1 − k2 i1
j
j
1
k1i 1 − k1 i1
j
2.15
k2i 1 − k2 ,
∀j ∈ Z .
i1
j
Note also that
k ≤ 1 for i 1, 2, for all j ∈ Z , ∃j0 j0 ε ∈ Z such that
1 i
j
ki ≤ ε < 1; for all j ≥ j0 for any given real constant ε ∈ 0, 1, an infinite subsequence
j
j
1
ji
ki }j ∈Z of { 1 ki }j∈Z which is monotone decreasing and limj → ∞ 1 ki 0
{ 1
i
for i 1, 2. Thus, if the sequences {kij } in 0, 1 for i 1, 2 are such that there is some finite
K ∈ R for which
j
1
j
k1i 1 − k1 i1
≤ lim sup
j →∞
j
k2i 1 − k2 i1
j j
1
k1i 1 − k1 i1
j
k2i 1 − k2 2.16
≤ K,
∀j ∈ Z
i1
according to 2.15 then if 2.11 holds, one gets from 2.11 that 2.12 holds for any given
ε ∈ R and some positive decreasing real sequence {εj }j∈Z and some nondecreasing positive
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real sequence {Kj }j∈Z subject to Kj ≤ Kj1 ≤ K, so that lim supj → ∞ Kj ≤ K, and εj1 ≤
εj ≤ ε which converges then to zero as j → ∞ since it has an infinite monotone decreasing
subsequence {εji }ji j∈Z for each j ∈ Z , satisfying to 0 < ji1 − ji < ∞ for any i, j ∈ Z ,
then limji → ∞ εji limj → ∞ εj 0. As a result, 2.13 follows, whose upper-bound is zero if
distA1 , A2 0.
Remark 2.3. Note that a simple comparison between 2.6 in Theorem 2.1 and 2.12 in
Corollary 2.2 concludes that K in 2.12 can be taken as small as 2 if the sequences {kij }j∈Z in
Corollary 2.2 are constant and equal to ki ∈ 0, 1 for i 1, 2. Therefore, a logic procedure
of accomplishing with 2.11 is to check for the existence of valid constants K possessing a
lower-bound 2.
If the constants k1 k2 i 1, 2 in Theorem 2.1, or the time-varying sequences
{k1j k2j }j∈Z for i 1, 2 of Corollary 2.2, are less than unity then the following result
holds.
Corollary 2.4. The following properties hold.
i Ti : A1 ∪ A2 → A1 ∪ A2 for i 1, 2 are both 2-cyclic contractive self-mappings with
time-varying real sequences {kij }j∈Z (i 1, 2) such that any element of the sum sequence
{k1j k2j }j∈Z is in 0, 1 for i 1, 2 with k ≤ maxj∈Z k1j k2j ∈ 0, 1 and k :
is some infinite subset of Z . Then,
minj∈Z k1j k2j ≥ 0 where Z
2 − ρk
distA1 , A2 ,
d T j x, y , T j x , y ≤ kj max d x, x d y, y 1−k
2.17
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z ,
where ρ : k /k if kk /
0 and k 0
0, ρ 1 if k k 0, and ρ 0 if k /
2 − ρk
distA1 , A2 ,
lim sup d T j x, y , T j x , y ≤
1−k
j →∞
∀x, y ∈ A1 , ∀x , y ∈ A2 .
2.18
ii Let Ti : A1 ∪ A2 → A1 ∪ A2 be both 2-cyclic nonexpansive and asymptotically contractive
self-mappings for i 1, 2 with the sequences {kij }j∈Z in 0, 1 for i 1, 2 and having
limits ki for i 1, 2 satisfying k : maxk1 , k 2 < 1/2, k : mink1 , k2 ≥ 0.
Then,
2 1 − ρk
distA1 , A2 ;
lim sup d T ji x, y , T ji x , y ≤
j→∞
1 − 2k
where ρ : mink1 , k2 / maxk1 , k2 ∈ 0, 1.
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.19
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Discrete Dynamics in Nature and Society
exists since the time-varying contraction
Proof. i First, note that the integer subset Z
sequences converge to real constants being less than unity. One gets from 2.5a for k :
mink1 , k2 ρk ≤ k < 1/2 where ρ k /k ≤ mink1 , k2 / maxk1 , k2 ∈ 0, 1
j−1 j j
j
≤ k max d x, x d y, y 2 − ρk distA1 , A2 d T x, y , T x , y
ki
2 − ρk 1 − k
≤ kj max d x, x d y, y 1−k
j
i0
distA1 , A2 2 − ρk
distA1 , A2 ,
≤ kj max d x, x d y, y 1−k
∀x, y ∈ A1 , ∀x , y ∈ A2 , ∀j ∈ Z
2.20
and 2.17 holds. Property i has been proven.
ii The proof of Property ii is close to that of Property i by using a close method to
that of the proof of Corollary 2.2ii:
j
d T j x, y , T j x , y ≤ 2k ε1j max d x, x d y, y
j j 2k εij
2 1 − ρk distA1 , A2 j
i1 i1
≤ 2k ε1j max d x, x d y, y
j 2 1 − ρk 1 − 2k
distA1 , A2 K j, 0
1 − 2k
j
≤ 2k ε0j max d x, x d y, y
2 1 − ρk
distA1 , A2 K j, 0 ,
1 − 2k
2.21
1/j−i
j
∈ R → 0 as
for all x, y ∈ A1 , for all x , y ∈ A2 , where εij : i1 k1j k2j − 2k
j> i → ∞, for all i ∈ Z0 , and {Kj, i}j>i∈Z ; are sequences of positive finite real constants
whose integer arguments j and i are related to the iterate of the left and right hand sides of
2.20, respectively, defined by
K j i, i
⎛
j ⎞
2 1 − ρk 1 − 2k
j
j
⎜ ⎟
⎟ distA1 , A2 .
: ⎜
−
2k εnj
⎝2 1 − ρk
⎠
1 − 2k
ni n1
2.22
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Since 0 ≤ k1j k2j ≤ 1; for all j ∈ Z0 then 0 ≤ 2k εij ≤ 1; for all i, j ∈ Z and since the
limit k1j k2j → k : maxk1 , k2 < 1/2 as j → ∞, {Kj, i}j>i∈Z is uniformly bounded and
limi → ∞ Kj i, i 0; for all j ∈ Z . Thus, combining 2.20 and 2.21, one gets
d T ji x, y , T ji x , y
≤ 2k ε0j
j
2 1 − ρk
distA1 , A2 K j i, i ,
max d T i x, T i x d T i y, T i y 1 − 2k
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.23
and then
lim sup
j>i → ∞, i → ∞
d T ji x, y , T ji x , y
j
≤ 2k ε0j lim sup max d T i x, T i x d T i y, T i y
lim sup
j>i → ∞, i → ∞
2 1 − ρk
i→∞
distA1 , A2 ,
1 − 2k
d T ji x, y , T ji x , y
2.24
lim sup d T ji x, y , T ji x , y
j →∞
≤
2 1 − ρk
1 − 2k
distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 .
Some Krasnoselskii-type fixed point results follow for the map T : A1 ∪ A2 × A1 ∪ A2 → X
defined trough 2-cyclic binary self-mappings for the case when A1 and A2 intersect.
Theorem 2.5. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume that A1 and
A2 are nonempty, convex, and closed subsets of X which intersect. Assume also that Ti : A1 ∪ A2 →
A1 ∪ A2 (i 1, 2) are both 2-cyclic contractive self-mappings and T : A1 ∪ A2 × A1 ∪ A2 → X
fulfilling T A1 ∩ A2 × A1 ∩ A2 ⊆ A1 ∩ A2 . Then, there is a unique fixed point z z1 z2 of
T : A1 ∪A2 ×A1 ∪A2 → X in A1 ∩A2 which satisfies T1 zT2 z z, where zi ∈ FixTi ⊆ A1 ∩A2
are also the respective unique fixed points of Ti : A1 ∪ A2 → A1 ∪ A2 (i 1, 2).
Proof. One gets from Theorem 2.1iii, 2.9
j
j
j
j
0 ←− d T j x, y , T j x , y ≤ d T1 x, T1 x d T2 y, T2 y −→ 0
j
j
j
j
⇒ d T1 x, T1 x −→ 0; d T2 y, T2 y −→ 0 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 as j → ∞,
2.25
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Discrete Dynamics in Nature and Society
∅ implies distA1 , A2 0. Since X is a Banach space then X, d is a complete
since A1 ∩ A2 /
metric space, since Ti : A1 ∪A2 → A1 ∪A2 i 1, 2 are 2-cyclic contractive self-mappings then
satisfying 1.1 with contraction constants ki ∈ 0, 1 and distA1 , A2 0 since A1 ∩ A2 / ∅
j, i 1, 2, and since T A1 ∩ A2 × A1 ∩ A2 ⊆ A1 ∩ A2 /
∅,
and subject Ti Aj ⊆ A for j, /
the unique limits below for Cauchy sequences exist
T j x, y −→ T z, z T1i z T2i z z ∈ A1 ∩ A2 ,
j
j
j
T1 x −→ T1 x −→ T1i z1 z1 ∈ A1 ∩ A2 ,
∀i ∈ Z as j → ∞,
j
T2 y −→ T2 x −→ T1i z2 ∈ A1 ∩ A2 ,
2.26
∀i ∈ Z as j → ∞, ∀x, y ∈ A1 ; ∀x , y ∈ A2 ,
where zi ∈ FixTi ⊆ A1 ∩ A2 i 1, 2 are unique in A1 ∩ A2 since A1 , A2 and then A1 ∩ A2 are nonempty, convex, and closed. It follows that z z1 z2 by making i → ∞ in T z, z T1i z T2i z z ∈ FixT ⊆ A1 ∩ A2 , for all i ∈ Z . Since z1 and z2 then z is unique so that
FixT {z}.
The following result follows from Corollary 2.2 and Theorem 2.5.
Corollary 2.6. Theorem 2.5 also holds if Ti : A1 ∪ A2 → A1 ∪ A2 (i 1, 2) are both 2-cyclic
asymptotically contractive self-mappings with respective time-varying contraction sequences being in
0, 1 with limits in 0, 1.
Theorem 2.7. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume that A1 and
A2 are nonempty, convex, and closed subsets of X, which intersect and have a convex union. Assume
that Ti : A1 ∪A2 → A1 ∪A2 (i 1, 2) are both 2-cyclic self-mappings, the first one being continuous,
the second one being contractive, and, furthermore, the mapping T : A1 ∪ A2 × A1 ∪ A2 →
X satisfies
T A1 ∪ A2 × A1 ∪ A2 ⊆ A1 ∪ A2 ,
T A1 ∩ A2 × A1 ∩ A2 ⊆ A1 ∩ A2 .
2.27
Then, there is a (in general, nonunique) fixed point z ∈ A1 ∪ A2 of T : A1 ∪ A2 × A1 ∪ A2 → X
which has the properties below:
T1i z T2i z z,
∀i ∈ Z ,
j
∃ lim T1 z z − y T1 z T2 z − y,
j →∞
2.28
where y T2i y ∈ FixT2 ≡ {y} ⊆ A1 ∩ A2 , for all i ∈ Z , is the unique fixed point of T2 :
A1 ∪ A2 → A1 ∪ A2 .
Proof. Note that the following properties hold
1 By hypothesis, A1 ∪A2 is a nonempty convex set which is, furthermore, closed since
A1 and A2 are both nonempty and closed.
2 A1 ∩A2 is nonempty, convex, and closed since A1 and A2 are nonempty, closed, and
convex with nonempty intersection.
Discrete Dynamics in Nature and Society
11
3 A1 ∪A2 ×A1 ∪A2 is invariant through the mapping T : A1 ∪A2 ×A1 ∪A2 → X
by hypothesis.
4 A1 ∪ A2 is a nonempty compact set which is trivially invariant under the 2-cyclic
continuous self-mapping T1 : A1 ∪ A2 → A1 ∪ A2 .
5 T2 : A1 ∪ A2 → A1 ∪ A2 is a 2-cyclic contraction which has then a unique fixed point
y ∈ FixT2 ≡ {y} ⊆ A1 ∩ A2 from Theorem 2.1 since A1 ∩ A2 is nonempty, convex,
and closed.
Thus, from the standard Krasnoselskii fixed point theorem 20, 21, under the above
Properties 1–4, it exists z ∈ FixT ⊆ A1 ∪ A2 which satisfies T1i z T2i z z T z, z; for all
j
i ∈ Z . Also, one gets from Property 5 that limj → ∞ T2 z T2 y y ∈ FixT2 ≡ {y} ⊆ A1 ∩ A2
are limits of Cauchy sequences so that
j
j
T1 z
j
T2 z
j
z T z, z → T1 z y as j → ∞ leading
to ∃ limj → ∞ T1 z z − y T1 z T2 z − y.
The following results follow from Krasnoselskii fixed point theorem extended to 2cyclic self-mappings since A1 ∪ A2 is a compact set 20.
Theorem 2.8. Theorem 2.7 holds if T2 : A1 ∪ A2 → A1 ∪ A2 is a 2-cyclic large contraction instead
of being a contraction.
Theorem 2.9. Theorem 2.7 holds if the hypothesis T A1 ∪ A2 × A1 ∪ A2 ⊆ A1 ∪ A2 is replaced
by x T1 x T2 y; ∀y ∈ A1 ∪ A2 ⇒ x ∈ A1 ∪ A2 .
Corollary 2.10. Theorem 2.7 also follows with the replacement of the 2-cyclic contractive selfmapping T2 : A1 ∪ A2 → A1 ∪ A2 by a 2-cyclic asymptotic contractive self-mapping T2 :
A1 ∪ A2 → A1 ∪ A2 for i 1, 2 with time-varying contraction sequences in 0, 1 with limits in
0, 1.
Corollary 2.10 follows by using Corollary 2.2 instead of Theorem 2.1. Also, Theorems
2.8 and 2.9 can be extended for T2 : A1 ∪ A2 → A1 ∪ A2 being asymptotically contractive
closely to the extension Corollary 2.10 to Theorem 2.7.
On the other hand, note that if A1 ∩ A2 ∅ then the convergence through the mapping
T : A1 ∪ A2 × A1 ∪ A2 → X to a fixed point under the conditions of Theorem 2.5, or
those of Corollary 2.6, cannot be achieved since fixed points, if any, are in A1 ∩ A2 since A1 ∩
A2 ∅. See Theorem 2.1, 2.6, Corollary 2.2, 2.13, Corollary 2.4, 2.18, or 2.19. It is not
either guaranteed the convergence to best proximity points because the guaranteed upperbounds for the limit superiors of distances of the iterates exceed the distance between the
adjacent subsets A1 and A2 since the lower upper-bounds of the above respective referred to
limit superiors are of the form λ distA1 , A2 for some real constant λ > 1 defined directly by
inspection of 2.6, 2.13, or 2.18, or 2.19. Then, the following result follows instead of
Theorem 2.5.
Theorem 2.11. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume also
that A1 and A2 are nonempty subsets of X, which do not intersect. Assume that Ti : A1 ∪ A2 →
A1 ∪ A2 (i 1, 2) are both 2-cyclic either contractive under Theorem 2.1 or under Corollary 2.4
12
Discrete Dynamics in Nature and Society
(i) (or asymptotically contractive under Corollary 2.2 or under Corollary 2.4(ii)) self-mappings, and,
furthermore,
T : A1 ∪ A2 × A1 ∪ A2 −→ A1 ∪ A2 .
2.29
Then, T j x, y is asymptotically permanent as j → ∞; for all x ∈ A1 , for all y ∈ A2 entering a
2 of clA1 ∪ A2 , where:
1 ∪ A
compact subset A
1 : {z ∈ A1 : distA1 , A2 ≤ distz, A2 ≤ λ distA1 , A2 };
A
2 : {z ∈ A2 : distA1 , A2 ≤ distz, A1 ≤ λ distA1 , A2 },
A
2.30
and λ > 1 is defined directly from 2.6, under Theorem 2.1; from 2.13, under Corollary 2.2, or from
either 2.18 or 2.19, under Corollary 2.4.
Proof. It follows from either Theorem 2.1, Corollary 2.2 or Corollary 2.4 since 2.29 holds.
Note that 2.29 which restricts the image of T : A1 ∪ A2 × A1 ∪ A2 → X, used in
Theorem 2.5 and in Corollary 2.6 is essential for the existence of the residual set where the
iterates T j x, y enter asymptotically as j → ∞. It is now proven how Corollary 2.4 may
be improved to obtain λ 1 in Theorem 2.11 so that the convergence of the iterates T j :
A1 ∪ A2 × A1 ∪ A2 → A1 ∪ A2 converge to a best proximity point if A1 and A2 are
nonempty, disjoint, convex, and closed. The part of Theorem 2.11 referred to the fulfilment
of Corollary 2.4 i.e., either the sum of both contraction constants or the sum of their limits if
they are time-varying sequences is less than unity is improved as follows.
Corollary 2.12. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume that A1
and A2 are nonempty, disjoint, convex, and closed subsets of X. Assume also that Ti : A1 ∪ A2 →
A1 ∪ A2 (i 1, 2) are both 2-cyclic either contractive under Corollary 2.4(i) with (or asymptotically
contractive under Corollary 2.4(ii)) self-mappings and, furthermore, (see 2.29). Then, any iterates
T j x, y, respectively, T j1 x, y converge to a best proximity point of either A1 , respectively, A2 , or
conversely, as j → ∞ for any given x ∈ A1 , y ∈ A2 .
Proof. Take some real constant k ∈ k1 k2 , 1 and note from 1.1 that 2.8 is modified as
follows:
d T x, y , T x , y ≤ k1 k2 max d x, x , d y, y 2 − k1 − k2 distA1 , A2 ≤ k max d x, x , d y, y 1 − k distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.31
which holds since
k − k1 − k2 max d x, x , d y, y − k − k1 − k2 distA1 , A2 ≥ 0
⇐⇒ max d x, x , d y, y ≥ distA1 , A2 , ∀x, y ∈ A1 , ∀x , y ∈ A2 .
2.32
Discrete Dynamics in Nature and Society
13
Thus,
d T j x, y , T j x , y ≤ k1 k2 max d x, x , d y, y 2 − k1 − k2 distA1 , A2 j
k
≤ k max d x, x , d y, y 1 − k distA1 , A2 j
0
kj max d x, x , d y, y 1 − kj distA1 , A2 ,
∀x, y ∈ A1 , ∀x , y ∈ A2 ,
2.33
so that the limit limj → ∞ dT j x, y, T j x , y distA1 , A2 exists modifying 2.18 in
i is a subset of the boundary
Corollary 2.4i. Then, Theorem 2.11 holds with λ 1 and A
of Ai containing the best proximity points of Ai to the set Ai1 i 1, 2 with A3 A1 . Thus,
since {T j x, y}j∈Z is a Cauchy sequence since X, d is complete, then T j x, y converges to
a best proximity point of A2 resp., A1 if j → ∞ and is odd resp., even. A close proof
leading to a similar modified limit follows by modifying 2.19 in Corollary 2.4ii by using
the limits ki of the time-varying asymptotically contractive sequences for i 1, 2, instead of
the constants ki , satisfying ki ∈ 0, 1/2 implying k1 k2 < 1 for k ∈ k1 k2 , 1.
Note that Corollary 2.4 and its referred to part of Corollary 2.12 also hold in a more
general version for limits ki ∈ 0, 1 with k1 k2 < 1 and k ∈ k1 k2 , 1.
Some illustrative examples follow.
Example 2.13. Consider the scalar differential equations
ẋi t ai xi t ri ,
xi 0 xi0 ,
i 1, 2,
2.34
with ai < 0 for i 1, 2. The solution trajectories are defined by contractive self-mappings
Ti : R → R for i 1, 2 as follows:
xi t, xi0 Ti xi0 t e
ai t
xi0 ri
t
e
−ai τ
dτ ,
i 1, 2; ∀xi0 ∈ R, ∀t ∈ R0 .
2.35
0
Both solutions converge to respective unique fixed points zi ri /|ai | as t → ∞ which are also
stable equilibrium points, for i 1, 2 since by using the Euclidean metric, it follows trivially
for any two initial conditions xi0 , xi0 i 1, 2
|xi t, xi0 − xi t, xi0 | ≤ e−|ai |t |xi0 − xi0 | → 0 as t → ∞ for i 1, 2,
2.36
and xi t, xi0 → zi with xi0 ∈ R as t → ∞ for i 1, 2. Note that the same property holds by
redefining the self-mappings T ih : R → R for i 1, 2 so as to pick up by successive iterates
14
Discrete Dynamics in Nature and Society
starting from initial conditions the sequence of points of the solutions {xi kh, xi0 }k∈Z0 for
some h ∈ R as
xi k 1h, xi0 T ih xi kh e
ai h
h
xi kh ri
e
−ai τ
dτ ,
i 1, 2; ∀xi0 ∈ R, ∀t ∈ R0 ,
0
2.37
which are both contractive since |xi k 1h, xi0 −xi k 1h, xi0 | ≤ e−|ai |h |xi kh−xi kh| →
0 as Z0 k → ∞ for i 1, 2 and xi kh, xi0 → zi with xi0 ∈ R as Z0 k → ∞ for i 1, 2.
Define the self-mapping T : R → R by T x, yt T1 xt T2 yt for any k ∈ Z ; for
k
all x, y ∈ R. Note that T x, yt → T z, z and T h x, y → T z, z Z0 k → ∞ for any
x, y ∈ R with z z1 z2 r1 /|a1 | r2 /|a2 |. Now, assume that the initial conditions are
restricted to fulfil the constraint |xi0 | ≤ Mi for i 1, 2. Consider a convex closed real interval
A B ≡ −M1 M2 r1 /|a1 | r2 /|a2 |, M1 M2 r1 /|a1 | r2 /|a2 |. Then, T x, yt and
k
T h x, y are in such an interval and converge to the fixed point z for any initial conditions
x, y ∈ A satisfying the constraint. Note that Ti A, Ti B ⊆ A ≡ B and T ih A, T ih B ⊆ A ≡ B
are trivially 2-cyclic contractive self-mappings from A to A Theorem 2.5.
Example 2.14. Now, assume the replacement r1 → r1 t in Example 2.13 where it exists the
!t
integral 0 e|a1 |τ r1 τdτ Oe|a1 |t ; for all t ∈ R0 . Then, the above results still hold with
the individual and combined resulting mappings still being contractive leading to respective
!t
unique fixed points z1 r 1 , z2 r2 /|a2 |, and z r 1 r2 /|a2 | with r 1 : 0 e−|a1 |t−τ r1 τdτ,
! t −|a |t−τ
subject to r1 : supt∈R 0 e 1
r1 τdτ < ∞, and A ≡ B −M1 M2 r1 r2 /|a2 |, M1 M2 r1 r2 /|a2 |.
Example 2.15. Consider self-mappings Ti : A ∪ B → A ∪ B i 1, 2 defined for any given
h, ε ∈ R , ai ∈ R− , and ri ∈ R i 1, 2 by
Ti xi kh xi k 1h, xi0 ,
xi k1h, xi0 ⎧
⎪
⎨xi k1h, xi0 e−|ai |h xi kh ri kh e|ai |h −1,
|ai |
⎪
⎩ε sign x k1h, x ,
i
i0
ri kh |ai | −1ki ki − e−|ai |h xi kh,
−1
h
e|ai |h
xi k1h, xi0 e
−|ai |h
xi kh ri
e
−ai τ
if |xi k1h, xi0 | ≥ ε,
otherwise,
dτ ;
i 1, 2; ∀x10 ∈ A; ∀x20 ∈ B, ∀t ∈ R0 ,
0
2.38
for all k ∈ Z0 , ki ∈ 0, 1 being given contraction constants for i 1, 2 for disjoint sets
A ≡ R2ε− ⊂ A ≡ Rε− , B ≡ Rε ≡ −A ≡ −Rε− : {z ∈ R : z ≥ ε} for some given ε ∈
R . The above equations are interpreted as the solution of the discretized versions of the
Discrete Dynamics in Nature and Society
15
differential equations 2.34 discussed in Examples 2.13 and 2.14 subject to saturated values
on the near boundaries of the real subintervals A and B. Note that Ti A ⊆ B and Ti B ⊆ A
for i 1, 2 so that Ti : A ∪ B → A ∪ B i 1, 2 are 2-cyclic self-mappings for which any
succession of iterates converges to best proximity points −2ε ∈ A and ε ∈ B for any bounded
initial conditions which restricts the iterates to be real successions on bounded convex disjoint
subsets of R. The composed self-mapping T : A ∪ B × A ∪ B → A ∪ B restricted to
T : A ∪ B × A ∪ B → A ∪ B by a domain restriction converges to best proximity points
−ε ∈ A and ε ∈ B Corollary 2.12. If now a1 ε x10 0 then A ∩ B {0} and z 0 is a fixed
point of T : A ∪ B × A ∪ B → A ∪ B. Note that the mapping T1 is continuous although it
is noncontractive while some Krasnoselskii-type theorems still apply.
The above examples are easily extended for nonscalar cases by using the same tools.
3. Results for Mappings Defined by P ≥ 2-Cyclic Self-Mappings
Define n : {1, 2, . . . , n} for any n ∈ Z . Then, the above results are easily extended to the
case of ≥2 p≥2 p-cyclic T : i∈p Ai → i∈p Ai ; for all ∈ n self-mappings which define
&n
a mapping T : i∈p Ai × i∈p Ai → X defined by T x1 , x2 , . . . , xn i1 Ti xi , for
j
any xi ∈ Aj and any given j ∈ p T : i∈p Ai × i∈p Ai → i∈p Ai . To facilitate the
exposition, assume that
1 T : i∈p Ai →
i∈p Ai are given p-cyclic noncontractive self-mappings for ∈
n1 ⊂ n where card n − card n2 ≥ card n1 ≥ 0,
2 T : i∈p Ai → i∈p Ai are p-cyclic contractive self-mappings for ∈ n2 : n\n1 ⊆ n,
where card n2 ≥ 1, with contraction constants k ∈ 0, 1; for all ∈ n2 .
That is, we allocate, with no loss of generality, the n-cyclic self-mappings which are
not contractive, while they are typically nonexpansive or continuous, all in the first strictly
ordered set of integers and those being contractive in the second strictly ordered one. The
identities of sets Aip ≡ Ai ; for all i ∈ p are assumed in all the necessary notations as associated
with p-cyclic self-mappings. It is also used that any p-cyclic nonexpansive self-mappings have
identical distances d : distAi , Ai1 > 0; for all i ∈ p between disjoint adjacent sets 6. The
following three results can be proven in a similar way as those counterparts of Section 2 for
p q 2.
Theorem 3.1. Assume that Ti : ∈p Ai → ∈p Ai for i ∈ n are p-cyclic nonexpansive for Part
(i), with at least one of them is being contractive p − 1 ≥ p2 ≥ 1 for Part (ii). Then, Theorem 2.1
holds with the subsequent replacements:
2 −→ n n2 ≥ 2;
diam A1 diam A2 −→
p
diam Ai ,
k1 k2 −→
i1
maxmink1 , k2 −→ maxmin ki : i ∈ p ,
p
ki ,
i1
distA1 , A2 −→ distAi , Ai1 , ∀i ∈ p ,
∀x, y ∈ A1 , ∀x , y ∈ A2 −→ ∀x, y ∈ Ai , ∀x , y ∈ Ai1 ; ∀i ∈ p .
3.1
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Discrete Dynamics in Nature and Society
Corollary 3.2. Assume that Ti : ∈p Ai → ∈p Ai for i ∈ n are p-cyclic asymptotically
contractive self-mappings. Then, Corollary 2.2 holds with the replacements:
ki ← kij ∈ 0, 1, i 1, 2, ∀j ∈ Z
∞←j
−→
ki ←− kij ∈ 0, 1, i ∈ p, ∀j ∈ Z
3.2
∞←j
and all further necessary replacements borrowed from Theorem 3.1.
Corollary 3.3. Assume that Ti : ∈p Ai → ∈p Ai for i ∈ n are all p-cyclic contractive for
Part (i) and p-cyclic nonexpansive and asymptotically contractive for Part (ii). Thus, Corollary 2.4
holds by replacing:
k1j k2j j∈Z −→
'
n
(
kij
k ≤ max k1j k2j
,
i1
j∈Z
j∈Z
k : min k1j k2j
j∈Z
⎛
−→ ⎝k ≥ max
j∈Z
⎛
'
(
n
⎝
kij
≥ 0 −→ k : min
j∈Z
i1
⎞
i1
⎞
(
⎠,
kij
j∈Z
⎠,
j∈Z
1
1
k : max k1 , k 2 <
−→ k : max ki : i ∈ n <
,
2
n
k : min k 1 , k 2 ≥ 0 −→ k : min ki : i ∈ n ≥ 0 ,
'
n
3.3
and all further necessary replacements borrowed from Theorem 3.1 or Corollary 2.2.
The following results based on Krasnoselskii theorems follow for fixed and best
proximity points in a Banach space endowed with a translation-invariant and homogeneous
metric. The results are a direct extension from the results for mappings T : A1 ∪ A2 ×
A1 ∪ A2 → A1 ∪ A2 built with mixed continuous and contractive 2-cyclic self-mappings
of Section 2 to mappings T : i∈p Ai × i∈p Ai → i∈p Ai built with p≥2-cyclic selfmappings, some of them being continuous while the remaining ones are contractive. The
underlying ideas are a a mapping which is a sum of continuous mappings is continuous,
b a set of contractive p-cyclic self-mappings for the same set of convex and closed subsets
of X has each a unique fixed point if the p subsets involved in the p-cyclic contractive selfmappings intersect. The sum of all those contractive self-mappings has a unique fixed point
which is the sum of the whole set of contractive self-mappings, c the combinations of
both results yields the existence of a fixed point from Krasnoselskii fixed point theorems
of T : i∈p Ai × i∈p Ai → i∈p Ai if all the subsets of the p-cyclic structure intersect. If
those sets do not intersect, there is a limiting best proximity point at any of the p subsets of X
for any p-cyclic iterates for initial points in between adjacent subsets.
Theorem 3.4. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume also that
Ai ; for all i ∈ p are nonempty, convex, and closed subsets of X with i∈p Ai also that
/ ∅. Assume
T : i∈p Ai → i∈p Ai ; for all ∈ n are p-cyclic contractive self-mappings fulfilling T i∈p Ai ×
&
i∈p Ai ⊆ i∈p Ai . Then, there is a unique fixed point z ni1 zi of T : i∈p Ai → X in i∈p Ai
Discrete Dynamics in Nature and Society
17
&
which satisfies ni1 Ti z z, where z ∈ FixT ⊆ i∈p Ai ; for all ∈ n are also the respective unique
fixed points of T : i∈p Ai → i∈p Ai ; for all ∈ n.
Theorem 3.4 above and Corollary 3.5 below extend, respectively, Theorem 2.5 and
Corollary 2.6 to the existence of fixed points for a mapping built from a set of n contractive,
respectively, asymptotically contractive, p≥2-cyclic self-mappings.
Corollary 3.5. Theorem 3.4 also holds if T : i∈p Ai →
i∈p Ai ; for all ∈ n are p-cyclic
asymptotically contractive self-mappings with respective time-varying contraction sequences being
in 0, 1 with limits in 0, 1.
Theorem 2.7 extends directly as follows
Theorem 3.6. Assume that X is a Banach space which has an associate complete metric space X, d
with the metric d : X × X → R0 being translation-invariant and homogeneous. Assume that A1 and
A2 are nonempty, convex, and closed subsets of X, which intersect and have convex union. Assume
that T : i∈p Ai → i∈p Ai ; for all ∈ n are p-cyclic self-mappings, of which n1 are continuous and
n2 are contractive subject to 1 ≤ n1 n − n2 and n2 ≥ 1. Assume, furthermore, that the mapping
T : i∈p Ai → X satisfies
⎛⎛
⎞ ⎛
⎞⎞
)
)
)
Ai ,
T ⎝⎝ Ai ⎠ × ⎝ Ai ⎠⎠ ⊆
i∈p
i∈p
⎛⎛
T ⎝⎝
i∈p
*
1
Ti z z,
∀i ∈ Z ,
j
*
⎞⎞
Ai ⎠ ⎠ ⊆
i∈p
i∈p
∃ lim Ti z z − y j →∞
⎛
Ai ⎠ × ⎝
i∈p
Then, there exists a (in general, nonunique) fixed point z ∈
X which has the properties below:
n
⎞
Ai of T : n
1
)
Ai .
3.4
i∈p
Ti z − y,
i∈p
Ai × ∀i ∈ n1 ,
i∈p
Ai →
3.5
&
&
where y nn1 1 Ti y nn1 1 y , y ∈ FixT ≡ {y } ⊆ i∈p Ai ; for all ∈ n2 , for all i ∈ Z ,
is the unique fixed point of the p-cyclic self-mapping T : i∈p Ai → i∈p Ai ; for all ∈ n.
Theorem 2.7 extends directly as follows
Theorem 3.7. Theorem 3.6 holds if T : i∈p Ai →
i∈p Ai ; for all ∈ n2 is a p-cyclic large
contraction instead of being a contraction, for all ∈ n.
Theorem 3.8. Theorem 3.6 holds if the hypothesis T i∈p Ai × i∈p Ai ⊆ i∈p Ai is replaced by
&
& 1 i
T x nn1 1 Ti y; ∀y ∈ i∈p Ai ⇒ x ∈ i∈p Ai .
x n1
One gets by extending Corollary 2.12 based on the ad hoc extensions of Theorem 2.20
and Corollary 2.4 for the composition of p-cyclic self-mappings.
Theorem 3.9. Assume that X is a uniformly convex Banach space which has an associate complete
metric space X, d with the metric d : X × X → R0 being translation-invariant and homogeneous.
Assume also that Ai for i ∈ p are nonempty, disjoint, convex and closed subsets of X. Furthermore,
assume that T : i∈p Ai × i∈p Ai → i∈p Ai . Then, the following properties hold.
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Discrete Dynamics in Nature and Society
i Assume that Ti : i∈p Ai → i∈p Ai are p-cyclic contractive self-mappings with time&
varying real sequences {kij }j∈Z ; for all i ∈ p such that the sum sequence { ni1 kij }j∈Z is
&
&
in 0, 1 with k ≤ maxj∈Z ni1 kij ∈ 0, 1 and k : minj∈Z ni1 kij ≥ 0 where Z
ji
is some infinite subset of Z . Then, any iterates T x, y converges as j → ∞ to a best
proximity point of Ai ≡ Ai−p for some integer p subject to 2p − i ≥ ≥ p − i − 1 for any
given x ∈ Aμ , y ∈ Aμ1 ; for all μ ∈ p.
ii Assume that Ti : i∈p Ai → i∈p Ai are p-cyclic asymptotically contractive self-mappings,
&
and, furthermore, with the sequences { ni1 kij }j∈Z in 0, 1 and having limits ki for i ∈ n
satisfying k : maxki : i ∈ n < 1/2, k : minki : i ∈ n ≥ 0. Then, any iterates
T ji x, y converges as j → ∞ to a best proximity point of Ai ≡ Ai−p for some integer
p subject to 2p − i ≥ ≥ p − i − 1 for any given x ∈ Aμ , y ∈ Aμ1 ; for all μ ∈ p.
Acknowledgments
The author is grateful to the Spanish Ministry of Education for its partial support of this paper
through Grant DPI2009-07197. He is also grateful to the Basque Government for its support
through Grants IT378-10 and SAIOTEK S-PE08UN15 and 09UN12. The author is grateful to
the reviewers for their constructive and useful comments.
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